Пространство L - множество элементов - векторов, которые обозначаются
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{\displaystyle {\vec {a}},{\vec {b}},\dots }
1.
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{\displaystyle {\textbf {1.}}{\vec {a}}+{\vec {b}}={\vec {b}}+{\vec {a}};}
2.
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{\displaystyle {\textbf {2.}}({\vec {a}}+{\vec {b}})+{\vec {c}}={\vec {a}}+({\vec {b}}+{\vec {c}});}
3.
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{\displaystyle {\textbf {3.}}\exists {\vec {0}}:\forall {\vec {a}},{\vec {a}}+{\vec {0}}={\vec {a}};}
4.
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{\displaystyle {\textbf {4.}}\forall {\vec {a}}\exists (-{\vec {a}}):{\vec {a}}+(-{\vec {a}})={\vec {0}};}
5.
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{\displaystyle {\textbf {5.}}\alpha (\beta {\vec {a}})=(\alpha \beta ){\vec {a}};}
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{\displaystyle {\textbf {6.}}1{\vec {a}}={\vec {a}};}
7.
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{\displaystyle {\textbf {7.}}\alpha ({\vec {a}}+{\vec {b}})=\alpha {\vec {a}}+\alpha {\vec {b}};}
8.
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{\displaystyle {\textbf {8.}}(\alpha +\beta ){\vec {a}}=\alpha {\vec {a}}+\beta {\vec {a}}.}
Определение. система
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{\displaystyle {\vec {e}}_{1},\dots ,{\vec {e}}_{n}}
⇔
{\displaystyle \Leftrightarrow }
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{\displaystyle \forall {\vec {x}}\in L:{\vec {x}}=\sum \limits _{j=1}^{n}x_{j}{\vec {e}}_{j}}
Теорема.
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{\displaystyle dim(L)=n\Leftrightarrow }
∃
{\displaystyle \exists }
Доказательство.
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{\displaystyle (\Rightarrow )}
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{\displaystyle dim(L)=n\Rightarrow }
∃
{\displaystyle \exists }
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{\displaystyle {\vec {e}}_{1},\dots ,{\vec {e}}_{n}}
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{\displaystyle \forall {\vec {x}}\in L:{\vec {x}}=\sum \limits _{j=1}^{n}x_{j}{\vec {e}}_{j}}
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{\displaystyle {\vec {e}}_{1},\dots ,{\vec {e}}_{n},{\vec {x}}}
∃
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{\displaystyle \exists \lambda _{1},\dots ,\lambda _{n},\mu }
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{\displaystyle \lambda _{1}{\vec {e}}_{1}+\lambda _{2}{\vec {e}}_{2}+\dots +\lambda _{n}{\vec {e}}_{n}+\mu {\vec {x}}={\vec {0}}.}
μ
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{\displaystyle \mu \not =0\Rightarrow {\vec {x}}=\sum \limits _{j=1}^{n}({\frac {-\lambda _{j}}{\mu }}){\vec {e}}_{j}}
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{\displaystyle (\Leftarrow )}
∃
{\displaystyle \exists }
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{\displaystyle {\vec {e}}_{1}\dots {\vec {e}}_{n}}
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{\displaystyle dim(L)=n}
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{\displaystyle {\vec {e}}_{1}\dots {\vec {e}}_{n}}
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{\displaystyle {\begin{matrix}&\lambda _{1}|{\vec {y}}_{1}=\alpha _{11}{\vec {e}}_{1}+\dots +\alpha _{1n}{\vec {e}}_{n}\\&\lambda _{2}|{\vec {y}}_{2}=\alpha _{21}{\vec {e}}_{1}+\dots +\alpha _{2n}{\vec {e}}_{n}\\&\dots \\&\lambda _{n+1}|{\vec {y}}_{n+1}=\alpha _{n+1,1}{\vec {e}}_{1}+\dots +\alpha _{n+1,n}{\vec {e}}_{n}\end{matrix}}}
A
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{\displaystyle A={\bigl \|}\alpha _{i,j}{\bigr \|}}
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{\displaystyle (n+1)}
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{\displaystyle n}
⇒
∃
λ
{\displaystyle \Rightarrow \exists \lambda }
λ
{\displaystyle \lambda }
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{\displaystyle {\vec {y}}}
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{\displaystyle \sum \limits _{j=1}^{n+1}\lambda _{j}{\vec {y}}_{j}=(\sum \limits _{j=1}^{n}\lambda _{j}\alpha _{j1}){\vec {e}}_{1}+(\sum \limits _{j=1}^{n}\lambda _{j}\alpha _{j2}){\vec {e}}_{2}+\dots +(\sum \limits _{j=1}^{n}\lambda _{j}\alpha _{jn}){\vec {e}}_{n}=0{\vec {e}}_{1}+\dots +0{\vec {e}}_{n}={\vec {0}}.\Rightarrow }
Следствие.
Все базисы в линейном пространстве L (
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{\displaystyle dim(L)=n}
Любая линейно независимая система из n векторов образует базис.
править
Линейное пространство L, dim(L) = n > 0,
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{\displaystyle {\vec {e}}_{1},\dots ,{\vec {e}}_{n}}
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∀
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L
∃
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{\displaystyle \Rightarrow \forall {\vec {x}}\in L\exists !x_{1},\dots ,x_{n}\in R^{1}(C^{1}):{\vec {x}}=\sum \limits _{j=1}^{n}x_{j}{\vec {e}}_{j}.{\vec {X}}={\begin{pmatrix}x_{1}\\\dots \\x_{n}\end{pmatrix}}.}
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{\displaystyle {\vec {f}}_{1},\dots ,{\vec {f}}_{n}\Rightarrow {\vec {f}}_{1}=\sum \limits _{j=1}^{n}t_{j1}{\vec {e}}_{j},\dots ,{\vec {f}}_{n}=\sum \limits _{j=1}^{n}t_{jn}{\vec {e}}_{j}.}
T
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{\displaystyle T={\bigl \|}t_{i,j}{\bigr \|}={\begin{pmatrix}t_{11}&t_{12}&\dots &t_{1n}\\\dots &\dots &\dots &\dots \\t_{n1}&t_{n2}&\dots &t_{nn}\end{pmatrix}}}
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{\displaystyle {\vec {e}}_{1},\dots ,{\vec {e}}_{n}}
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{\displaystyle {\vec {f}}_{1},\dots ,{\vec {f}}_{n}}
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{\displaystyle {\vec {x}}=\sum \limits _{j=1}^{n}x_{j}{\vec {e}}_{j}=\sum \limits _{j=1}^{n}x'_{j}{\vec {f}}_{j}=\sum \limits _{i=1}^{n}x'_{i}(\sum \limits _{j=1}^{n}t_{ij}{\vec {e}}_{j});x_{j}=\sum \limits _{i=1}^{n}t_{ji}x'_{i}}
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{\displaystyle {\vec {X}}={\begin{pmatrix}x_{1}\\\dots \\x_{n}\end{pmatrix}},{\vec {X}}'={\begin{pmatrix}x'_{1}\\\dots \\x'_{n}\end{pmatrix}}.}
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{\displaystyle {\vec {X}}=T{\vec {X}}',{\vec {X}}'=T^{-1}{\vec {X}}}
